If α < 0, it is the closure of the complement of a disk of radius −1/α.

Then an edge of the alpha-shape is drawn between two members of the finite point set whenever there exists a generalized disk of radius 1/α containing the entire point set and which has the property that the two points lie on its boundary.

If α = 0, then the alpha-shape associated with the finite point set is its ordinary convex hull.

Alpha shapes are closely related to alpha complexes, subcomplexes of the Delaunay triangulation of the point set.

Each edge or triangle of the Delaunay triangulation may be associated with a characteristic radius, the radius of the smallest empty circle containing the edge or triangle. For each real numberα, the α-complex of the given set of points is the simplicial complex formed by the set of edges and triangles whose radii are at most 1/α.

The union of the edges and triangles in the α-complex forms a shape closely resembling the α-shape; however it differs in that it has polygonal edges rather than edges formed from arcs of circles. More specifically, Edelsbrunner (1995) showed that the two shapes are homotopy equivalent. (In this later work, Edelsbrunner used the name "α-shape" to refer to the union of the cells in the α-complex, and instead called the related curvilinear shape an α-body.)

This technique can be employed to reconstruct a Fermi surface from the electronic Bloch spectral function evaluated at the Fermi level, as obtained from the Green function in a generalised ab-initio study of the problem. The Fermi surface is then defined as the set of reciprocal space points within the first Brillouin zone, where the signal is highest.
The definition has the advantage of covering also cases of various forms of disorder.